From the planar Ising model to quasiconformal mappings

Abstract: We identify the scaling limit of full-plane Kadanoff-Ceva fermions on generic, non-degenerate $s$-embeddings. In this broad setting, the scaling limits are described in terms of solutions to conjugate Beltrami equations with prescribed singularities. For the underlying Ising model, this leads to the scaling limit of the energy-energy correlations and reveals a connection between the scaling limits of (near-)critical planar Ising models and quasiconformal mappings. For grids approximating bounded domains in the complex plane, we establish, in the scaling regime, the conformal covariance of the energy density on critical doubly periodic graphs. We complement this result with an analogous statement in the case where the limiting conformal structure generates a maximal surface $(z,\vartheta)$ in Minkowski space $\mathbb{R}^{(2,1)}$. All scaling factors obtained are local and expressed in terms of the geometry of the embedding, even in situations where they vary drastically from one region to another. These results confirm the predictions of Chelkak and highlight that the scaling limits of generic (near-)critical Ising models naturally live on a substantially richer conformal structure than the classical Euclidean one.